I've got stuck on an induction problem, any help would be great

A(1)=√5 A(n+1)=√5*a(n)

The terms of the sequence are: √5, √(5√5), √(5(√5(√5))),....

Use induction to show that for natural number n, 0<A(n)<5

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- May 8th 2010, 06:57 AMiwish123Induction
I've got stuck on an induction problem, any help would be great

A(1)=√5 A(n+1)=√5*a(n)

The terms of the sequence are: √5, √(5√5), √(5(√5(√5))),....

Use induction to show that for natural number n, 0<A(n)<5 - May 8th 2010, 08:09 AMArchie Meade
Hi iwish123,

Show by induction that

If

then

hence

As

then

So, being < 5__causes__all subsequent terms of the sequence to be < 5 also.

A non-inductive proof is as follows...

The index is a geometric series, first term=1, common ratio = 0.5 (for the part in brackets)

so the index of 5 is 1.

Hence cannot reach 5 as a finite number of terms will not reach the sum to infinity as all the terms are positive. - May 8th 2010, 08:21 AMiwish123
Thanks, thats a great help

How would I go about proving that the sequence is convergent, through showing its an increasing sequence and bounded? I assume that I can say it is bounded as its been shown that the sequence is less than 5. - May 8th 2010, 08:31 AMArchie Meade